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On elliptic systems with Sobolev critical exponent
Quasisymmetric geometry of the Cantor circles as the Julia sets of rational maps
1. | School of Mathematical Sciences, Fudan University, Shanghai 200433, China |
2. | Department of Mathematics, Nanjing University, Nanjing 210093, China |
3. | Department of Mathematics, Zhejiang University, Hangzhou 310027, China |
References:
[1] |
A. F. Beardon, Iteration of Rational Functions, Graduate Texts in Mathematics, 132, Springer-Verlag, New York, 1991.
doi: 10.1007/978-1-4612-4422-6. |
[2] |
M. Bonk, Uniformization of Sierpiński carpets in the plane, Invent. Math., 186 (2011), 559-665.
doi: 10.1007/s00222-011-0325-8. |
[3] |
M. Bonk, M. Lyubich and S. Merenkov, Quasisymmetries of Sierpiński carpet Julia sets, preprint,, , ().
|
[4] |
M. Bourdon, Immeubles hyperboliques, dimension conforme et rigidité de Mostow, (French) [Hyperbolic buildings, conformal dimension and Mostow rigidity], Geom. Funct. Anal., 7 (1997), 245-268.
doi: 10.1007/PL00001619. |
[5] |
M. Bourdon and H. Pajot, Quasi-conformal geometry and hyperbolic geometry, in Rigidity in Dynamics and Geometry, Springer, Berlin, 2002, 1-17. |
[6] |
G. Cui, Dynamics of rational maps, topology, deformation and bifurcation, Preprint, May, 2002 (early version: Geometrically finite rational maps with given combinatorics, 1997). |
[7] |
R. L. Devaney, D. Look and D. Uminsky, The escape trichotomy for singularly perturbed rational maps, Indiana Univ. Math. J., 54 (2005), 1621-1634.
doi: 10.1512/iumj.2005.54.2615. |
[8] |
A. Douady and J. H. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. Éc Norm. Sup., 18 (1985), 287-343. |
[9] |
M. Gromov, Hyperbolic groups, in Essays in Group Theory, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987, 75-263.
doi: 10.1007/978-1-4613-9586-7_3. |
[10] |
P. Haïssinsky, Géométrie quasiconforme, analyse au bord des espaces métriques hyperboliques et rigidités, Astérisque, 326 (2009), 321-362. |
[11] |
P. Haïssinsky and K. Pilgrim, Quasisymmetrically inequivalent hyperbolic Julia sets, Rev. Mat. Iberoam., 28 (2012), 1025-1034.
doi: 10.4171/RMI/701. |
[12] |
J. Heinonen, Lectures on Analysis on Metric Spaces, Universitext, Springer-Verlag, New York, 2001.
doi: 10.1007/978-1-4613-0131-8. |
[13] |
M. Kapovich and B. Kleiner, Hyperbolic groups with low-dimensional boundary, Ann. Sci. Éc Norm. Sup., 33 (2000), 647-669.
doi: 10.1016/S0012-9593(00)01049-1. |
[14] |
B. Kleiner, The asymptotic geometry of negatively curved spaces: Uniformization, geometrization and rigidity, in International Congress of Mathematicians, II, Eur. Math. Soc., Zürich, 2006, 743-768. |
[15] |
O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, Springer-Verlag, Berlin, Heidelberg, New York, 1973. |
[16] |
C. McMullen, Automorphisms of rational maps, in Holomorphic Functions and Moduli I, Math. Sci. Res. Inst. Publ., 10, Springer, New York, 1988, 31-60.
doi: 10.1007/978-1-4613-9602-4_3. |
[17] |
J. Milnor, Dynamics in One Complex Variable: Third Edition, Annals of Mathematics Studies, 160, Princeton Univ. Press, Princeton, NJ, 2006. |
[18] |
K. Pilgrim and L. Tan, Rational maps with disconnected Julia sets, Astérisque, 261 (2000), 349-383. |
[19] |
W. Qiu, X. Wang and Y. Yin, Dynamics of McMullen maps, Adv. Math., 229 (2012), 2525-2577.
doi: 10.1016/j.aim.2011.12.026. |
[20] |
W. Qiu, F. Yang and Y. Yin, Rational maps whose Julia sets are Cantor circles, Ergod. Th. & Dynam. Sys., 35 (2015), 499-529.
doi: 10.1017/etds.2013.53. |
[21] |
N. Steinmetz, On the dynamics of the McMullen family $R(z)=z^m+\lambda/z^l$, Conform. Geom. Dyn., 10 (2006), 159-183.
doi: 10.1090/S1088-4173-06-00149-4. |
[22] |
L. Tan and Y. Yin, Local connectivity of the Julia sets for geometrically finite rational maps, Sci. China Ser. A, 39 (1996), 39-47. |
show all references
References:
[1] |
A. F. Beardon, Iteration of Rational Functions, Graduate Texts in Mathematics, 132, Springer-Verlag, New York, 1991.
doi: 10.1007/978-1-4612-4422-6. |
[2] |
M. Bonk, Uniformization of Sierpiński carpets in the plane, Invent. Math., 186 (2011), 559-665.
doi: 10.1007/s00222-011-0325-8. |
[3] |
M. Bonk, M. Lyubich and S. Merenkov, Quasisymmetries of Sierpiński carpet Julia sets, preprint,, , ().
|
[4] |
M. Bourdon, Immeubles hyperboliques, dimension conforme et rigidité de Mostow, (French) [Hyperbolic buildings, conformal dimension and Mostow rigidity], Geom. Funct. Anal., 7 (1997), 245-268.
doi: 10.1007/PL00001619. |
[5] |
M. Bourdon and H. Pajot, Quasi-conformal geometry and hyperbolic geometry, in Rigidity in Dynamics and Geometry, Springer, Berlin, 2002, 1-17. |
[6] |
G. Cui, Dynamics of rational maps, topology, deformation and bifurcation, Preprint, May, 2002 (early version: Geometrically finite rational maps with given combinatorics, 1997). |
[7] |
R. L. Devaney, D. Look and D. Uminsky, The escape trichotomy for singularly perturbed rational maps, Indiana Univ. Math. J., 54 (2005), 1621-1634.
doi: 10.1512/iumj.2005.54.2615. |
[8] |
A. Douady and J. H. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. Éc Norm. Sup., 18 (1985), 287-343. |
[9] |
M. Gromov, Hyperbolic groups, in Essays in Group Theory, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987, 75-263.
doi: 10.1007/978-1-4613-9586-7_3. |
[10] |
P. Haïssinsky, Géométrie quasiconforme, analyse au bord des espaces métriques hyperboliques et rigidités, Astérisque, 326 (2009), 321-362. |
[11] |
P. Haïssinsky and K. Pilgrim, Quasisymmetrically inequivalent hyperbolic Julia sets, Rev. Mat. Iberoam., 28 (2012), 1025-1034.
doi: 10.4171/RMI/701. |
[12] |
J. Heinonen, Lectures on Analysis on Metric Spaces, Universitext, Springer-Verlag, New York, 2001.
doi: 10.1007/978-1-4613-0131-8. |
[13] |
M. Kapovich and B. Kleiner, Hyperbolic groups with low-dimensional boundary, Ann. Sci. Éc Norm. Sup., 33 (2000), 647-669.
doi: 10.1016/S0012-9593(00)01049-1. |
[14] |
B. Kleiner, The asymptotic geometry of negatively curved spaces: Uniformization, geometrization and rigidity, in International Congress of Mathematicians, II, Eur. Math. Soc., Zürich, 2006, 743-768. |
[15] |
O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, Springer-Verlag, Berlin, Heidelberg, New York, 1973. |
[16] |
C. McMullen, Automorphisms of rational maps, in Holomorphic Functions and Moduli I, Math. Sci. Res. Inst. Publ., 10, Springer, New York, 1988, 31-60.
doi: 10.1007/978-1-4613-9602-4_3. |
[17] |
J. Milnor, Dynamics in One Complex Variable: Third Edition, Annals of Mathematics Studies, 160, Princeton Univ. Press, Princeton, NJ, 2006. |
[18] |
K. Pilgrim and L. Tan, Rational maps with disconnected Julia sets, Astérisque, 261 (2000), 349-383. |
[19] |
W. Qiu, X. Wang and Y. Yin, Dynamics of McMullen maps, Adv. Math., 229 (2012), 2525-2577.
doi: 10.1016/j.aim.2011.12.026. |
[20] |
W. Qiu, F. Yang and Y. Yin, Rational maps whose Julia sets are Cantor circles, Ergod. Th. & Dynam. Sys., 35 (2015), 499-529.
doi: 10.1017/etds.2013.53. |
[21] |
N. Steinmetz, On the dynamics of the McMullen family $R(z)=z^m+\lambda/z^l$, Conform. Geom. Dyn., 10 (2006), 159-183.
doi: 10.1090/S1088-4173-06-00149-4. |
[22] |
L. Tan and Y. Yin, Local connectivity of the Julia sets for geometrically finite rational maps, Sci. China Ser. A, 39 (1996), 39-47. |
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